function [J grad] = nnCostFunction(nn_params, ...
                                   input_layer_size, ...
                                   hidden_layer_size, ...
                                   num_labels, ...
                                   X, y, lambda)
%NNCOSTFUNCTION Implements the neural network cost function for a two layer
%neural network which performs classification
%   [J grad] = NNCOSTFUNCTON(nn_params, hidden_layer_size, num_labels, ...
%   X, y, lambda) computes the cost and gradient of the neural network. The
%   parameters for the neural network are "unrolled" into the vector
%   nn_params and need to be converted back into the weight matrices. 
% 
%   The returned parameter grad should be a "unrolled" vector of the
%   partial derivatives of the neural network.
%

% Reshape nn_params back into the parameters Theta1 and Theta2, the weight matrices
% for our 2 layer neural network
Theta1 = reshape(nn_params(1:hidden_layer_size * (input_layer_size + 1)), ...
                 hidden_layer_size, (input_layer_size + 1));

Theta2 = reshape(nn_params((1 + (hidden_layer_size * (input_layer_size + 1))):end), ...
                 num_labels, (hidden_layer_size + 1));

% Setup some useful variables
m = size(X, 1);
X = [ones(m, 1) X]; %update X

         
% You need to return the following variables correctly 
J = 0;
Theta1_grad = zeros(size(Theta1));
Theta2_grad = zeros(size(Theta2));

% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the code by working through the
%               following parts.
%
% Part 1: Feedforward the neural network and return the cost in the
%         variable J. After implementing Part 1, you can verify that your
%         cost function computation is correct by verifying the cost
%         computed in ex4.m
for i = 1:m
    z2 = Theta1 * (X(i, :)');
    a2 = sigmoid(z2);
    a2 = [1; a2];
    z3 = Theta2 * a2;
    a3 = sigmoid(z3);
    y_real = zeros(num_labels, 1);
    y_real(y(i)) = 1;
    J = J + y_real'*log(a3) + (1 .- y_real)'*log(1 .- a3);
end

J = J * (-1.0 / m);

    
%
% Part 2: Implement the backpropagation algorithm to compute the gradients
%         Theta1_grad and Theta2_grad. You should return the partial derivatives of
%         the cost function with respect to Theta1 and Theta2 in Theta1_grad and
%         Theta2_grad, respectively. After implementing Part 2, you can check
%         that your implementation is correct by running checkNNGradients
%
%         Note: The vector y passed into the function is a vector of labels
%               containing values from 1..K. You need to map this vector into a 
%               binary vector of 1's and 0's to be used with the neural network
%               cost function.
%
%         Hint: We recommend implementing backpropagation using a for-loop
%               over the training examples if you are implementing it for the 
%               first time.

% Part 3: Implement regularization with the cost function and gradients.
%
%         Hint: You can implement this around the code for
%               backpropagation. That is, you can compute the gradients for
%               the regularization separately and then add them to Theta1_grad
%               and Theta2_grad from Part 2.
%
regulValue = 0;
s1 = input_layer_size;
s2 = hidden_layer_size;
s3 = num_labels;
Theta1New = Theta1(:, 2:end);
Theta2New = Theta2(:, 2:end);

for j = 1:s2
    regulValue += Theta1New(j, :) * (Theta1New(j, :)');
end

for j = 1:s3
    regulValue += Theta2New(j,:) * (Theta2New(j, :)');
end

regulValue *= lambda / (2 * m);

J += regulValue;

%init Weight space of partial derivative
Delta1 = zeros(s2, s1+1);
Delta2 = zeros(s3, s2+1);

for i = 1 : m
    a1 = X(i, :); %To get sample(i)'s data
    
    %cal forward propagation
    z2 = Theta1 * (a1');
    a2 = sigmoid(z2);
    a2 = [1; a2];
    z3 = Theta2 * a2;
    a3 = sigmoid(z3);

    %calculate the difference between each layer
    y_real = zeros(num_labels, 1);
    y_real(y(i)) = 1;   %the y(output value)'s real value
    diff3 = a3 - y_real; %the difference of the third layer
    diff2 = (Theta2(:, 2:end)')*diff3 .* a2(2:end) .*(1 .- a2(2:end));

    %Accumulate the weight of the second layer of partial derivative
    Delta2(:, :) = Delta2 .+ diff3 * (a2');
    Delta1(:, :) = Delta1 .+ diff2 * a1;
end

Theta1_grad(:, :) = (1.0/m) .* Delta1 + (lambda/m) .* Theta1;
Theta1_grad(:, 1) = Theta1_grad(:, 1) - (lambda/m) .* Theta1(:, 1);
Theta2_grad(:, :) = (1.0/m) .* Delta2 + (lambda/m) .* Theta2;
Theta2_grad(:, 1) = Theta2_grad(:, 1) - (lambda/m) .* Theta2(:, 1);


% -------------------------------------------------------------

% =========================================================================

% Unroll gradients
grad = [Theta1_grad(:) ; Theta2_grad(:)];


end
